Examples 5.2.7(a):

Consider the collection of sets
(0, 1/j) for all j &gt 0. What is the intersection of
all of these sets ?

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The intersection of all intervals (0, 1/j) is empty. To see
this, take any real number x. If
x 0
it is not in any of the intervals (0, 1/j), and hence not
in their intersection. If x &gt 0, then there exists an
integer N such that 0 &lt 1 / N &lt x. But then
x is not in the set (0, 1 / N) and therefore
x is not in the intersection. Therefore, the intersection is
empty.

Note that this is an intersection of 'nested' sets, that is sets that
are decreasing: every 'next' set is a subset of its predecessor.